Divergent Trajectories in Arithmetic Homogeneous Spaces of Rational Rank Two

Abstract

Let $G$ be a real algebraic group defined over $\mathbb{Q}$, $\Gamma$ be an arithmetic subgroup of $G$, and $T$ be a maximal $\mathbb{R}$-split torus. A trajectory in $G/\Gamma$ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which account for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in $T$, we study the existence of non-obvious divergent trajectories under its action in $G/\Gamma$. We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that $\mbox{rank}_{\mathbb{Q}}G=\mbox{rank}_{\mathbb{R}}G=2$.